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Classical Probability Distributions
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Radu T. Trîmbiţaş
UBB
1st Semester 2010-2011
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
1 / 46
Bernoulli
Experiment: n independent trials, each is a succes or a failure, the
probability of succes remains constant from trial to trial – Bernoulli process
Xk :
0 1
q p
,
k = 1, . . . , n, q = 1
p
E (Xk ) = 0 q + 1 p = p
V (Xk ) = E (Xk2 )
Radu T. Trîmbiţaş (UBB)
E (Xk )2 = 02 q + 12 p
Classical Probability Distributions
p 2 = pq
1st Semester 2010-2011
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Binomial
(Bernoulli, 1713)
A r.v. X has a binomial distribution if its mass function is
p (x ) =
(xn )p x q n
0,
x,
x = 0, 1, . . . , n
otherwise
X denotes the number of successes in n Bernoulli trials
X can be written as a sum of n independent Bernoulli r.v.
X = X1 + X2 +
+ Xn
So,
E (X ) = np
V (X ) = npq
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
3 / 46
Applications
Examples
1
Plot pdf and cdf of a binomial distribution with p = 0.2 and n = 10.
2
A Quality Assurance inspector tests 200 circuit boards a day. If 2% of
the boards have defects, what is the probability that the inspector will
…nd no defective boards on any given day? What is the most likely
number of defective boards the inspector will …nd?
3
Suppose that a lot of 300 electrical fuses contains 5% defectives. If a
sample of …ve fuses is tested, …nd the probability of observing at least
one defective.
4
Experience has shown that 30%of all persons a- icted by a certain
illness recover. A drug company has developed a new medication.
Ten people with the illness were selected at random and injected with
the medication; nine recovered shortly thereafter. Suppose that the
medication was absolutely worthless. What is the probability that at
least nine of ten injected with the medication will recover?
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
4 / 46
Geometric distribution
The number of trials to achieve the …rst success
p (x ) =
qx
1 p,
0,
x = 1, 2, . . .
otherwise
Mean and variance
∞
E (X ) =
∑ kpqk
1
k =1
V (X ) =
=
1
p
q
p2
The geometric distribution is useful for modelling the runs of
consecutive successes (or failures) in repeated independent trials of a
system. The geometric distribution models the number of successes
before one failure in an independent succession of tests where each
test results in success or failure.
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
5 / 46
Examples
1
Plot a pdf and a cdf of a geometric distribution with p = 0.5.
2
Suppose you toss a fair coin repeatedly. If the coin lands face up
(heads), that is a success. What is the probability of observing
exactly three tails before getting a heads? What is the probability of
observing three or fewer tails before getting a heads?
3
40% of the assembled ink-jet printers are rejected at the inspection
station. Find the probability that the …rst acceptable ink-jet printer is
the thirs one inspected.
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
6 / 46
Negative binomial distribution
X has a negative binomial distribution with parameters r 2 N and
p 2 (0, 1) if its mass function is
f (x jr , p ) =
r +x
x
1
pr qx ,
x 2N
It can be thought of as modelling the total number of failures that
occured before the nth success
rq
E (X ) =
p
rq
V (X ) = 2
p
Sometimes X is considered to be the number of trials to amass a
total o r successes
P (X = n ) =
Radu T. Trîmbiţaş (UBB)
n
r
1 r n
p q
1
r
,
Classical Probability Distributions
n = r , r + 1, . . .
1st Semester 2010-2011
7 / 46
Applications
Examples
1
Plot the pdf and the cdf of a negative binomial distribution for r = 3
and p = 0.5
2
A geological study indicates that an exploratory oil well drilled in a
particular region should strike oil with the probability 0.2. Find the
probability that the oil strike comes on the …fth well drilled and on
the …rst …ve wells drilled.
3
40% of the assembled ink-jet printers are rejected at the inspection
station. Find the probability that the …rst printer inspected is the
second acceptable printer.
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
8 / 46
Poisson Distribution
A random variable X is said to have a Poisson distribution with
parameter > 0 i¤ its mass function is
f (x j λ ) =
λx
e
x!
λ
,
x 2 N.
The Poisson distribution is appropriate for applications that involve
counting the number of times a random event occurs in a given
amount of time, distance, area, etc.
Mean, variance
E (X ) = V (X ) = λ.
As Poisson showed, the Poisson distribution is the limiting case of a
binomial distribution where n approaches in…nity and p goes to zero
while np = λ .
First used to model deaths from the kicks of horses in the Prussian
Army.
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
9 / 46
Applications I
1
Plot the pdf and the cdf of a Poisson distribution for λ = 5.
2
Suppose that a random system of police patrol is devised so that a
patrol o¢ cer may visit a given beat location Y = 0, 1, 2, . . . times per
half-hour period, with each location being visited an average of once
per time period. Assume that Y possesses, approximately, a Poisson
probability distribution. Calculate the probability that the patrol
o¢ cer will miss a given location during a half-hour period. What is
the probability that it will be visited once? Twice? At least once?
3
A computer hard disk manufacturer has observed that ‡aws occur
randomly in the manufacturing process at the average rate of two
‡aws in a 4 Gb hard disk and has found this rate to be acceptable.
What is the probability that a disk will be manufactured with no
defects?
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
10 / 46
Applications II
4
Consider a Quality Assurance department that performs random tests
of individual hard disks. Their policy is to shut down the
manufacturing process if an inspector …nds more than four bad
sectors on a disk. What is the probability of shutting down the
process if the mean number of bad sectors (λ) is two?
5
A repair person is “beeped” each time there is a call service. The
number of beeps is Poisson with λ = 2 per hour. Find the probability
of three beeps in the next hour, and the probability of more than
three beeps i an 1-hour period.
6
The lead time demand in an inventory system is the accumulation of
demand for an item at which an order is placed until the order is
received
T
L=
∑ Di ,
i =1
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
11 / 46
Applications III
where L is the lead-time demand, Di is the demand during the ith
time period, and T is the number of time periods during the lead
time. Both Di and T may be random variables. An inventory
manager desires that the probability of a stockout not exceed a
certain fraction (e.g. 5%) during the lead time. The roerder point is
the level of inventory at which a new order is placed. Assume that the
lead-time demand is poisson distributed with a mean of λ = 10 units
and that a 95% protection of a stockout is desired. That is, …nd the
smallest x such that the probability that the lead-time demand does
not exceed x is 95%.
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
12 / 46
Hypergeometric distribution
Parameters (they have physical interpretations): N is the size of the
population, m is the number of items with the desired characteristic
in the population, m N. n is the number of samples drawn.
Sampling “without replacement”.
The pmf
f (x jN, m, n) =
m
x
N
n
N
n
m
x
,
x = 0, 1, . . . , n, x
K, n
x
M
mean, variance
nm
.
N
nm (N n)(N m )
V (X ) =
.
N 2 (N 1)
E (X ) =
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
13 / 46
Applications
1
Plot the pdf and the cdf of an experiment taking 20 samples from a
group of 1000 where there are 50 items of the desired type.
2
Suppose you have a lot of 100 ‡oppy disks and you know that 20 of
them are defective. What is the probability of drawing 0 through 5
defective ‡oppy disks if you select 10 at random? What is the
probability of drawing zero to two defective ‡oppies if you select 10 at
random?
3
Suppose you are the Quality Assurance manager for a hard disk
manufacturer. The production line turns out disks in batches of
1,000. You want to sample 50 disks from each batch to see if they
have defects. You want to accept 99% there are no more than 10
defective disks in the batch. What is the maximum number of
defective disks should you allow in your sample of 50?
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
14 / 46
Uniform distribution
X has a U (a, b ) distribution it its pdf is
8
< 1
, x 2 [a, b ]
f (x ja, b ) =
b a
: 0,
otherwise
cdf
mean, variance
8
0,
x <a
>
< x a
F (x ja, b ) =
, a x <b
> b a
:
1,
x b
a+b
2
(b a )2
V (X ) =
12
E (X ) =
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
15 / 46
Applications
1
Plot the graphs of pdf and cdf of a U [0, 1] distribution.
2
What is the probability that an observation from a uniform
distribution with a = 1 and b = 1 will be less than 0.75?
3
What is the 99th percentile of the uniform distribution between -1
and 1?
4
A bus arrives every 20 minutes at a speci…ed stop beginning at 6:40
A.M. and continuing until 8:40 A.M. A certain passenger does not
know the schedule, but arrives randomly (uniformly distributed)
between 7:00 A.M. and 7:30 A.M. every morning. What is the
probability that the passenger waits more than 5 minutes for a bus.
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
16 / 46
Normal distribution
X has a N (µ, σ2 ) distribution, µ 2 R, σ > 0 if its pdf is
f (x jµ, σ2 ) =
1
p e
σ 2π
cdf
(x µ )2
2σ2
,
x 2R
Z
x
(t µ )2
1
p
e 2σ2 dt
σ 2π ∞
2
for µ = 0, σ = 1, standard normal distribution
Laplace’s function – cdf of standard normal
F (x jµ, σ2 ) =
Φ (x ) =
1
p
σ 2π
Z x
∞
e
t2
2
dt
mean, variance
E (X ) = µ
V (X ) = σ 2
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
17 / 46
Gauss
Pierre Simon Laplace
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
18 / 46
Central Limit Theorem
Theorem
Let X1 , X2 , . . . be a sequence of independent and identically distributed
random variable having a …nite mean µ and a …nite variance σ2 . Then
lim P
n !∞
Radu T. Trîmbiţaş (UBB)
X1 +
X
pn
σ n
nµ
<x
Classical Probability Distributions
= Φ (x )
1st Semester 2010-2011
19 / 46
Applications
1
2
Plot the pdf and the cdf of the standard normal distribution
Plot on the same graph the pdf of N (0, 12 ), N (0, 22 ), N (0, 32 ). Do
the same for the cdf.
3
Check the 3σ rule for a given normal distribution.
4
Find an interval that contains 95% of the values from a standard
normal distribution. Note this interval is not the only such interval,
but it is the shortest. Find another, longer.
5
Lead-time demand X for an item is approximated by a N (25, 9)
distribution. It is desired to compute the value for led time that will
be exceeded only 5% of the time, that is …nd x0 such that
P (X > x0 ) = 0.05.
6
The time to pass through a queue to begin self-service at a cafeteria
has been found to be N (15, 9). Compute the probability that an
arriving customer waits between 14 and 17 minutes.
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
20 / 46
Lognormal distribution I
If Y is N (µ, σ2 ) then X = e Y has a lognormal distribution with
parameters µ and σ (X is log N (µ, σ2 ))
pdf
f (x jµ, σ) =
1
p
σx 2π
cdf
exp
"
#
(ln x µ)2
,
2σ2
(ln x µ)
p
=Φ
σ 2
unde Φ este funcţia lui Laplace, iar
F (x jµ, σ) =
1
erfc
2
erfc(x ) = 1
Radu T. Trîmbiţaş (UBB)
2
erf (x ) = p
π
Classical Probability Distributions
Z ∞
e
x > 0.
ln x µ
σ
t2
dt
x
1st Semester 2010-2011
21 / 46
Lognormal distribution II
Mean, variance
E (X ) = e µ + σ
2 /2
V (X ) = e 2µ+σ
2
eσ
2
1
If the mean and the variance are known to be µL and σL2 , respectively,
then
1
0
µL
A
µ = ln @ q
2
2
µL + σL
σ2 = ln
Radu T. Trîmbiţaş (UBB)
µ2L + σL2
µ2L
Classical Probability Distributions
.
1st Semester 2010-2011
22 / 46
Lognormal distribution III
median, mode
Mode (X ) = e µ
σ2
Median(X ) = e µ
A variable might be modeled as log-normal if it can be thought of as
the multiplicative product of many independent random variables
each of which is positive. For example, in …nance, a long-term
discount factor can be derived from the product of short-term
discount factors. In wireless communication, the attenuation caused
by shadowing or slow fading from random objects is often assumed to
be log-normally distributed. Economists often model the distribution
of income using a lognormal distribution.
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
23 / 46
Plot of lognormal pdfs
Figure: Plot of three lognormal pdfs for µ = 1 and σ = 1/2, 1, 2
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
24 / 46
Applications
1
Suppose the income of a family of four in the United States follows a
lognormal distribution with µ = ln(20, 000$) and σ2 = 1. Plot the
income density. What is the probability that the income be larger
than 60000$.
2
The rate of return on a volatile investment is modeled as having a
lognormal distribution with mean 20% and standard deviation 5%.
Compute the parameters for the lognormal distribution.
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
25 / 46
Beta distribution I
A random variable X is said to have a beta distribution with
parameters α, β > 0 if and only if its density function is
f (x jα, β) =
where
8
< xα
:
B (a, b ) =
(1 x ) β
B (α, β)
0,
1
xa
x )b
1
Z 1
1
(1
x 2 (0, 1)
otherwise
1
dx
0
is the Euler’s beta function.
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
26 / 46
Beta distribution II
mean, variance
E (X ) =
V (X ) =
α
α+β
αβ
2
( α + β ) ( α + β + 1)
The graphs of beta density exhibit a large variety of shapes for various
values of the two parameters α and β (see Figure below). Beta
distribution can be de…ned on an arbitrary interval (c, d ): if
x 2 (c, d ) then x = (x c )/(d c ) de…nes a new variable such
that x 2 (0, 1).
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
27 / 46
Beta distribution III
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
28 / 46
Applications I
1
A gasoline wholesale distributor has bulk storage tanks that hold …xed
supplies and are …lled every Monday. Of interest to the wholesaler is
the proportion of this supply that is sold during the week. Over many
weeks of observation, the distributor that this proportion could be
modelled by a beta distribution with α = 4 and β = 2. Find the
probability that the wholesaler will sent at least 90% of her stock in a
given week.
2
Rule of succession. A classic application of the beta distribution is
the rule of succession, introduced in the 18th century by Pierre-Simon
Laplace in the course of treating the sunrise problem. It states that,
given s successes in n conditionally independent Bernoulli trials with
1
probability p, that p should be estimated as ns +
+2 . This estimate may
be regarded as the expected value of the posterior distribution over p,
namely Beta(s + 1, n s + 1), which is given by Bayes’rule if one
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
29 / 46
Applications II
assumes a uniform prior over p (i.e., Beta(1, 1)) and then observes
that p generated s successes in n trials.
3
Task duration modeling. The beta distribution can be used to
model events which are constrained to take place within an interval
de…ned by a minimum and maximum value. For this reason, the beta
distribution — along with the triangular distribution — is used
extensively in PERT, critical path method (CPM) and other project
management / control systems to describe the time to completion of
a task. In project management, shorthand computations are widely
used to estimate the mean and standard deviation of the beta
distribution:
E (X ) =
σ (X ) =
Radu T. Trîmbiţaş (UBB)
a + 4b + c
q 6
V (X ) =
Classical Probability Distributions
c
a
6
1st Semester 2010-2011
30 / 46
Applications III
where a is the minimum, c is the maximum, and b is the most likely
value. Using this set of approximations is known as three-point
estimation and are exact only for particular values of α and β,
speci…cally when:
p
2
α=3
p
β = 3+ 2
or vice versa. These are notably poor approximations for most other
beta distributions exhibiting average errors of 40% in the mean and
54% in the variance.
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
31 / 46
Triangular distribution I
X has a triangular distribution with parameters a, b, c, a
its pdf is
8
2(x a )
>
>
, x 2 [a, b ]
>
>
< (b a)(c a)
2(c x )
f (x ja, b, c ) =
, x 2 (b, c ]
>
>
> (c b )(c a)
>
:
0,
elsewhere
b
c if
mean, mode
a+b+c
3
Mode = b = 3E (X )
E (X ) =
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
(a + c )
1st Semester 2010-2011
32 / 46
Triangular distribution II
cdf
8
>
>
>
>
>
>
<
0
x
a
2
(x a )
x 2 (a, b ]
(b a)(c a)
F (x ja, b, c ) =
>
(c x )2
>
>
1
x 2 (b, c ]
>
>
(c b )(c a)
>
:
1
x >c
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
33 / 46
Applications I
1
A central processor unit requirements, for programs that will execute,
have a triangular distribution with a = 0.05 ms, b = 1.1 ms, and
c = 6.5 ms. Find the probability that the CPU requirements for a
random program is 2.5 ms or less.
2
Implement MATLAB functions for the pdf, cdf, icdf of a triangular
distribution.
3
An electronic sensor evaluates the quality of memory chips, rejecting
those thai fail. Upon demand the sensor will give the minimum and
maximum number of rejects over the past 24 hours. The mean is also
given. Without further information, the quality control department
has assumed thet the number of rejected chips can be approximated
by a triangular distribution. The current dump data indicates that the
minimum number of rejected chips during any hour was zero, the
maximum was 10, and the mean was 4. Find a, b, and c and the
median.
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
34 / 46
Applications II
4
Find conditions on a, b, c such that mean, mode and median be
equal.
5
Find the variance and the median of a Triang (a, b, c ) r.v.
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
35 / 46
Exponential distribution I
X is said to be exponential distributed with parameter λ > 0 if its pdf
is
λe λx , x 0
f (x ) =
0,
otherwise
cdf
F (x ) =
1
0
e
λx ,
x <0
x 0
mean, variance, median
1
λ
1
V (X ) = 2
λ
ln 2
Me (X ) =
λ
E (X ) =
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
36 / 46
Exponential distribution II
variant (in statistical packages)
( 1
x
e λ, x 0
f (x ) =
λ
0,
otherwise
E (X ) = λ,
V (X ) = λ2
Exponential distribution models interarrival times when arrivals are
completely random and to model service time that are highly variable.
In this instances λ is a rate: arrivals per hour or services per minute
model the lifetime of a component that fails catastrophically
(instantaneously), luch as a light bulb; λ is the failure rate.
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
37 / 46
Graphs
Figure: Exponential pdfs for several values of λ
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Classical Probability Distributions
1st Semester 2010-2011
38 / 46
Properties I
Memoryless property:
P (X > s + t jX > s ) = P (X > t )
Proof
P (X > s + t jX > s ) =
P (X > s + t )
e λ (s +t )
=e
=
P (X > s )
e λs
λt
= P (X > t )
The exponential distributions and the geometric distributions are the
only memoryless probability distributions.
An exponential r.v. remains exponential when multiplied by a
constant
λ
x
P (cX
x) = P X
= 1 e cx
c
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Classical Probability Distributions
1st Semester 2010-2011
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Properties II
if X1 , . . . , Xn are e.i.r.v. with parameters λ1 , . . . , λn then
M = min(X1 , . . . , Xn ) is exponential with ∑i λi
P
Xj = min Xi jM > t
i
= P Xj
t = min(Xi
t )jM > t
= P Xj
t = min(Xi
t )jXi > t, i = 1, n
i
i
= P Xj = min Xi
i
(we used the memoryless property)
P (M > t ) = P (Xi > t, i = 1, . . . , n) = e
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
∑ni=1 λi t
1st Semester 2010-2011
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Properties III
Probability that Xj is the smallest (limits 0, ∞)
P (Xj = M ) =
=
Z
Z
Z
P (Xj = M jXj = t ) λj e
λj t
dt
P (Xi > t, i 6= j jXj = t ) λj e
λj t
dt
P (Xi > t, i 6= j ) λj e λj t dt
!
Z
=
∏ e λi t λj e λj t dt
=
= λj
Radu T. Trîmbiţaş (UBB)
Z
i 6 =j
e
∑i λi t dt
=
Classical Probability Distributions
λj
∑i λi
1st Semester 2010-2011
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Applications
ln 2
λ .
Prove this fact this
1
The median of the exponential distribution is
fact.
2
What is the probability that an exponential random variable will be
less than or equal to 1/E (X )?
3
Let the lifetime of light bulbs be exponentially distributed with
β = 700 hours. What is the median lifetime of a bulb?
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
42 / 46
Weibull distribution I
X has a Weibull distribution with parameters ν 2 R, α > 0, β > 0 if
its pdf is
8
"
#
β 1
>
x ν β
< β x ν
exp
,
x ν
f (x jν, α, β) =
α
α
α
>
:
0,
otherwise
cdf
F (x ) =
8
>
<
>
: 1
exp
" 0,
x
ν
α
β
#
x<ν
, x
ν
ν - location parameter, α - scale parameter, β - shape parameter
ν = 0 two parameter Weibull
ν = 0, β = 1, exponential with parameter λ =
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1
α
1st Semester 2010-2011
43 / 46
Weibull distribution II
mean
E (X ) = ν + αΓ
"
V (X ) = α
2
Γ
1
+1
β
2
+1
β
Γ
1
+1
β
2
#
an appropriate analytical tool for modeling the breaking strength of
materials. Current usage also includes reliability and lifetime
modeling. The Weibull distribution is more ‡exible than the
exponential for these purposes.
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
44 / 46
Graphs
Figure: Weibul pdfs for ν = 0 and α =
Radu T. Trîmbiţaş (UBB)
1
2
and various values for β
Classical Probability Distributions
1st Semester 2010-2011
45 / 46
Applications
1
Reproduce Figure 3. Plot cdfs for the same distributions.
2
Find the formula for the median of a Weibull distribution.
3
The time to failure for a component screen is known to have a Weibull
distribution with ν = 0, β = 1/3, and α = 200 hours. Find the mean,
the variance and the probability that a unit fails before 2000 hours.
4
The time it takes for an aircraft to land and clear the runaway at a
major international airport has a Weibull distribution with ν = 1.34
minutes, β = 0.5 and α = 0.04 minutes. Find the probability that an
incoming airpalne will take more than 1.5 minutes to land and cler
the runaway.
Radu T. Trîmbiţaş (UBB)
Classical Probability Distributions
1st Semester 2010-2011
46 / 46